How to Calculate Induced EMF and Self-Inductance in a Long Solenoid?

[libelec]

Homework Statement

A variable current I(t) = 2Acos(100Hz*t) is passed through a long, thin solenoid of R = 2,5 cm and 900 spirals per meter in length. Calculate the induced EMF inside the solenoid and the self-inductance coefficient L.

Homework Equations

Magnetic field inside a long, thin solenoid: $$\vec B = {\mu _0}\eta I{\rm{ \hat k}}$$, where $$\eta$$ is the spiral density (900 spirals per meter).

Induced EMF inside a solenoid: $$\varepsilon = \frac{{{\mu _0}{N^2}IS}}{L}$$, where S is the transversal suface.

The Attempt at a Solution

My problem is the lack of data. I don't have the length of the solenoid, so what I calculated remains a function of L. This is what I did:

1) I calculate the flux of B through one spiral: $$\Phi = {\mu _0}\eta I(t).{\pi ^2}R$$.

2) I multiply that by N (number of spirals), to get the total flux through the solenoid: $${\Phi _T} = N{\mu _0}\eta I(t).{\pi ^2}R$$.

3) Since it changes with time, because I changes with time, I derive the total flux to get the induced EMF: $$\varepsilon = - \frac{{d{\Phi _T}}}{{dt}} = - N{\mu _0}\eta {\pi ^2}R\frac{{dI}}{{dt}} = N{\mu _0}\eta {\pi ^2}R.2A.100Hz\sin (100Hz*t)$$

Then I can't calculate it. I'm missing the total number of spirals N, or the length L, such that $$\eta$$ = N/L.

What can I do to find the induced EMF. The problem asks for a numerical solution (in function of t).

 

[gabbagabbahey]

The way I see it, you have two options:

(1) Assume the solenoid is long enough to be considered infinite and calculate the emf and inductance per unit length (the total emf and inductance of an infinite solenoid is, not surprisingly, infinite)

(2) Assume the solenoid has some finite length $d$ and calculate the total emf and inductance using the exact field of a finite solenoid (not easy to calculate!)

Personally, I would choose option (1); both for ease of calculation, and since I strongly suspect that is what the questioner is looking for (since it is a very common textbook problem).

 

[libelec]

OK, thank you.